The saddle point equation (REF ) has been used in the second equality. This result counts how many times \(\bar{J}\)  penetrates the loops, and it reduces to the well-known expression in the continuous theory[1]}[2]} when the lines don't intersect. When the lines intersect, however, fractional linking numbers will be allowed, but the result is still well-defined, which is different from the continuous theory. Several examples are shown below:
